We consider linear plants controlled by dynamic output feedback which are subjected to blockdiagonal stochastic parameter perturbations. The stability radii of these systems are characterized, and it is shown that, for real data, the real and the complex stability radii coincide. A corresponding result does not hold in the deterministic case, even for perturbations of single-output feedback type. Ln a second part of the paper we study the problem of optimizing the stability radius by dynamic linear output feedback. Necessary and sufficient conditions are derived for the existence of a compensator which achieves a suboptimal stability radius. These conditions consist of a parametrized Riccati equation, a parametrized Liapunov inequality, a coupling inequality, and a number of linear matrix inequalities (one for each disturbance term). The corresponding problem in the deterministic case, the optimal mu-synthesis problem, is still unsolved.